What are real world applications of trigonometry?
Why do I have to learn trigonometry?
Where will I ever use this in real life?

Anyone who has taught trigonometry has heard these questions. Our answers usually involve Engineering applications, finding heights of buildings, surveying or maybe stock market or business cycles. Below are some examples of applications of trigonometry in situations where the use of trigonometry might be unexpected. These questions were sent to Quandaries and Queries, the question and answer service at Math Central.

The first quesion comes from Kent, a business owner and artisan who creates and sells metal sculptures. He is looking for the dimensions required to make a three-dimensional, five pointed star.

I am looking for a formula that will give me a layout for a 3 dimensional 5 pointed star. I want to form it out of sheet metal, using 5 polygons and soldering them at the apex. Can you please help me with this? I would like to be able to give the formula the height of the star from the bottom two points to the top point and also how deep the star is. Thank you very much!

Judi solved his problem and her solution appears in the Quandaries and Queries database. (She also has instructions on how to make a paper star.) Kent made his star and sent us the pictures below of the completed project in place,

and a closeup of the star.

The second question comes from Claudia, a frustrated homeowner who knew that her property assessment was being calculated inappropriately and wanted to know the correct area of her lot.

I own a piece of property that I need to know the square feet for assessment purposes. The figure they came up with is wrong. They measured from one point to another and halved the sums but that means I own the cul de sac and we don't. My lot is 55 feet wide and one side is 108.96 feet and the other side is 146.04 that extends all the way to a circle. The front of the lot on the cul de sac is stated on the survey like this. 78.21 feet where R=40 feet. This large arc is taken off the size of our land. How many square feet is our lot.

Her lot is in the shape of a rectangle with a circular piece removed, as in the diagram below.

Harley used trigonometry to calculate the area of her lot and show that this area had indeed been miscalculated.

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